Approximate solutions to ordinary differential equations using least squares support vector machines.

Area of Science:

• Numerical analysis
• Machine learning
• Applied mathematics

Background:

• Ordinary differential equations (ODEs) are fundamental in modeling complex systems.
• Existing numerical methods for ODEs face challenges with stiffness, nonstiffness, and singularities.
• Accurate and efficient solution techniques for ODEs are crucial across scientific disciplines.

Purpose of the Study:

• To introduce a new computational method for solving linear and nonlinear ODEs.
• To leverage least squares support vector machines (LS-SVMs) for approximating ODE solutions.
• To demonstrate the versatility of the proposed LS-SVM approach for diverse ODE problems.

Main Methods:

• The proposed method utilizes least squares support vector machines (LS-SVMs) to derive approximate solutions.
• LS-SVM parameters are optimized by minimizing a defined error function.
• Solutions are obtained by solving systems of linear or nonlinear equations, depending on the ODE type.

Main Results:

• The LS-SVM approach yields approximate solutions in a closed-form expression.
• The method is effective for mildly stiff, nonstiff, and singular ODEs.
• Numerical experiments confirm the efficiency and superiority of the LS-SVM method compared to existing approaches.

Conclusions:

• The proposed LS-SVM based method offers an efficient and accurate technique for solving a wide range of ODEs.
• This approach provides a robust alternative for problems involving initial and boundary conditions.
• The findings highlight the potential of machine learning techniques in advancing scientific computation for differential equations.